In 1903 Whittaker [1]} showed that with the help of proposed by
him function, it is possible to express some special functions like Bessel
functions, parabolic cylinder functions, error functions, incomplete gamma
functions, and logarithm and cosine integrals. Two Whittaker functions are
applied today and they are defined by using the Kummer confluent
hypergeometric function [2]}, [3]}:
\(\begin{array}{l}\displaystyle \mathrm {M}_{\kappa ,\mu }\left( x\right) =x^{\mu -1/2}e^{-x/2}\,_{1}F_{1}\left( \left.\begin{array}{c}\mu -\kappa +\frac{1}{2} \\1+2\mu \end{array}\right|x\right) , \\\displaystyle \mathrm {W}_{\kappa ,\mu }\left( x\right) =\frac{\Gamma \left( -2\mu \right)}{\Gamma \left( \frac{1}{2}-k-\mu \right) }\mathrm {M}_{\kappa ,\mu }\left(x\right) +\frac{\Gamma \left( 2\mu \right) }{\Gamma \left( \frac{1}{2}-k+\mu \right) }\mathrm {M}_{\kappa ,-\mu }\left( x\right) .\end{array}\) 
